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(Observe that, if s is close to e in G, then the image of H under the mapping
xisxs"1x~1 is close to e.)

8. Let G be a locally compact metrizable group and H a closed normal subgroup of G,
such that G/H is not discrete and has no small subgroups. Show that there exists a
continuous homoniorphism/: R ~^G such that the composition R->G->G/H is
nontrivial (see Section 12.9, Problem 7).


We shall adhere to the conventions of (5.1), so that all vector spaces under
consideration have as field of scalars either the real field R or the complex
field C.

If E is a vector space over R (resp. C), a topology on E is said to be com-
patible with the vector space structure if the mappings (x, y)i~+x + y of
E x E into E, and (1, x) h~> fa of R x E (resp. C x E) into E are continuous. A
vector space endowed with a topology compatible with its vector space
structure is called a topological vector space. Since  x = (I)*, the condi-
tions above imply that the topology is a fortiori compatible with the additive
group structure of E, and all the notions and properties established for
topological groups in the preceding sections will apply in particular to topolo-
gical vector spaces (provided of course that these properties are transcribed
into additive notation).

An isomorphism of a topological vector space E onto a topological vector
space F is a linear bijection of E onto F which is a homeomorphism.

If || A: || is a norm on a vector space E, the topology on E defined by the
distance \\x  y\\ is compatible with the vector space structure of E (5.1.5) and
therefore makes E a topological vector space. Two equivalent norms (5.6)
define the same topology.

In a topological vector space E, the translations x\~*x + a and the
homotheties x\-+fa(A ^ 0) are homeomorphisms of E onto E, the inverse
mappings being, respectively, the translation x\~-+x  a and the homothety

In a vector space E over R (resp, C) a set M is said to be balanced if,
for each x e M and each scalar A such that |A| ^ 1, we have Ax e M (in other
words, if AM c M whenever \A\  1). The set M is said to be absorbing if, for
each x e E, there exists a real number a > 0 such that /be e M whenever
|/l| <jl a. Every absorbing subset of E generates the vector space E. For a
balanced subset M of E to be absorbing, it is enough that for each x e E there
should exist a scalar A ^ 0 such that fa e M.(b)   Deduce from (a) that if HI, H2 are two normal subgroups of a topological group